# Probability and Venn Diagrams

Students learn how to calculate the probability of an event happening using sample space and Venn Diagrams.  Later, learning progresses on to calculating the probability of a conditional event using tree diagrams.

This unit takes place in Year 10 Term 1 and follows on from working with fractions and mixed numbers.

##### Prerequisite Knowledge
• Compare and order fractions whose denominators are all multiples of the same number
• Identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths
• Add and subtract fractions with the same denominator and denominators that are multiples of the same number
##### Success Criteria
• Record describe and analyse the frequency of outcomes of probability experiments using tables and frequency trees
• Apply ideas of randomness, fairness and equally likely events to calculate expected outcomes of multiple future experiments
• Relate relative expected frequencies to theoretical probability,
• Apply the property that the probabilities of an exhaustive set of outcomes sum to one; apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one
• Understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size
• Enumerate sets and combinations of sets systematically, using tables, grids, Venn diagrams and tree diagrams
• Construct theoretical possibility spaces for single and combined experiments with equally likely outcomes and use these to calculate theoretical probabilities
• Calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions
##### Key Concepts
• When writing probabilities as a fraction using the probability scale to show equivalences with the keywords
• Discuss the effect of bias and sample size when comparing theoretical and experimental probabilities.
• Use the random function on a calculator or spreadsheet to demonstrate simple randomisation.
• When listing the outcomes of combined events ensure students use a logical and systematic method.
• Branches on a probability tree have a sum of one as they are mutually exclusive.
• Conditional probability is where the outcome of a future event is dependent on the outcome of a previous event.
##### Common Misconceptions
• Writing probabilities as a ratio is a common misconception.
• When creating Venn diagrams students often forget to place the remaining events outside the circles.
• When listing permutations of combined events students often repeat events when they do not use a logical and systematic method.
• Students often have difficulty completing Venn diagrams involving 3 intersecting circles.

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